In his book 15 Minutes: General Curtis LeMay and the Countdown to Nuclear Annihilation, L. Douglas Keeney mentions in a couple spots that SAC suspected the Soviets might secretly place high-yield nuclear weapons at the bottoms of major US harbors, with a view to detonating them at the start of a nuclear war.
Is there any evidence that the Soviets had planned to (or attempted to) do just that?
(I realize that the Soviets apparently never fielded 100-MT devices as outlined in the scenario, but even a 15 or 20-megaton bomb would cause a hell of a lot of damage.)
Conversely, is there evidence that SAC expected the Soviets to do that because they planned to do something similar to Soviet ports?
It was assumed, not just by SAC and the American military, but by the British as well, that Soviet mini-subs could be used to place nuclear weapons in harbours and estuaries. This fear was most evident in the late 1940s, and early 1950s. In response to this, the RN ordered 20 seaward defence boats, of the 'Ford' class. However, such preparations were unnecessary - the Soviets had no midget submarines capable of delivering nuclear weapons at the time. They did have several submersibles, which were used to support spetsnaz forces, but these were not able to carry a nuclear device. Several midget submarines were, however, launched in the late 1980s.
As to your second question, the US had no capabilities for this. Their only midget submarine was the X-1, launched in 1954, to test the USN's ability to defend against attacks of this type . Also, it's worth noting that SAC wouldn't have had control over such plans - America's seaborne nuclear weapons were, and are, entirely under control of the Navy. However, the RN did have both plans for this and the capability to carry it out. Four midgets of the 'Stickleback'-class, slight modifications of the WW2 X and XE types, were built. They were supposedly built for countermeasures training, but they were actually meant to lay nuclear weapons off Leningrad. The weapons used would either be the Blue Danube, with a 20 kiloton yield, or the Red Beard, with a yield of 15-25 kilotons. One member of this class, Stickleback, can be seen at the Imperial War Museum at Duxford.
Sources:
Rebuilding the Royal Navy: Warship Design since 1945, David K. Brown, George Moore, Seaforth, 2012
The Naval Institute Guide to the Soviet Navy, Norman Polmar, United States Naval Institute, 1991